Linear oscillatory as well as transient and steady shear measurements for micellar solutions of dodecyltrimethylammonium, bromide (DTAB) and sodium salicylate (NaSal) as a function of the salt-to-surfactant concentration ratio (C-SALT/CDTAB) are presented. Our results indicate that, for molar ratios of salt to surfactant (C-SALT/CDTAB) smaller than 1, the micellar solutions follow closely Maxwell behavior with a single relaxation time, that is, they are in the fast-breaking regime, and exhibit shear banding above a critical shear rate. When this ratio is greater than 1, micellar solutions behave as semidilute polymer solutions with a spectrum of relaxation times and no stress plateau is observed in steady shear. The results are analyzed with the Granek-Cates model (for linear response) and with a simple model that consists of the upper convected Maxwell constitutive equation coupled to a kinetic equation to account for the breaking and reformation of the micelles (nonlinear behavior). The stress plateau and the critical shear rates are determined from an extended irreversible thermodynamic analysis.